This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A070910 #66 Feb 16 2025 08:32:46
%S A070910 2,2,7,9,5,8,5,3,0,2,3,3,6,0,6,7,2,6,7,4,3,7,2,0,4,4,4,0,8,1,1,5,3,3,
%T A070910 3,5,3,2,8,5,8,4,1,1,0,2,7,8,5,4,5,9,0,5,4,0,7,0,8,3,9,7,5,1,6,6,4,3,
%U A070910 0,5,3,4,3,2,3,2,6,7,6,3,4,2,7,2,9,5,1,7,0,8,8,5,5,6,4,8,5,8,9,8,9,8,4,5,9
%N A070910 Decimal expansion of BesselI(0,2).
%D A070910 Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 2, equation 2:5:5 at page 20.
%H A070910 Michael Penn, <a href="https://www.youtube.com/watch?v=-UhFu0g9740">An exponential trigonometric integral.</a>, YouTube video, 2020.
%H A070910 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FactorialSums.html">Factorial Sums</a>.
%H A070910 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ModifiedBesselFunctionoftheFirstKind.html">Modified Bessel Function of the First Kind</a>.
%F A070910 Equals Sum_{k>=0} 1/k!^2.
%F A070910 From _Peter Bala_, Aug 19 2013: (Start)
%F A070910 Continued fraction expansion: 1/(1 - 1/(2 - 1/(5 - 4/(10 - 9/(17 - ... - (n-1)^2/(n^2+1 - ...)))))). See A006040. Cf. A096789.
%F A070910 This continued fraction is the particular case k = 0 of the result BesselI(k,2) = Sum_{n = 0..oo} 1/(n!*(n+k)!) = 1/(k! - k!/((k+2) - (k+1)/((2*k+5) - 2*(k+2)/((3*k+10) - ... - n*(n+k)/(((n+1)*(n+k+1)+1) - ...))))). See the remarks in A099597 for a sketch of the proof. (End)
%F A070910 From _Amiram Eldar_, May 29 2021: (Start)
%F A070910 Equals (1/e^2) * Sum_{k>=0} binomial(2*k,k)/k! = e^2 * Sum_{k>=0} (-1)^k*binomial(2*k,k)/k!.
%F A070910 Equal (1/(2*Pi)) * Integral_{x=0..2*Pi} exp(2*sin(x)) dx. (End)
%F A070910 Equals BesselJ(0,2*i). - _Jianing Song_, Sep 18 2021
%e A070910 2.2795853023360672674372044408115333532858411...
%t A070910 RealDigits[ BesselI[0, 2], 10, 110] [[1]] (* _Robert G. Wilson v_, Jul 09 2004 *)
%t A070910 (* Or *) RealDigits[ Sum[ 1/(n!n!), {n, 0, Infinity}], 10, 110][[1]]
%o A070910 (PARI) besseli(0,2) \\ _Charles R Greathouse IV_, Feb 19 2014
%Y A070910 Cf. A096789, A070913 (continued fraction), A006040.
%Y A070910 Bessel function values: A334380 (J(0,1)), A334383 (J(0,sqrt(2))), A091681 (J(0,2)), A197036 (I(0,1)), A334381 (I(0,sqrt(2))), this sequence (I(0,2)).
%K A070910 cons,easy,nonn
%O A070910 1,1
%A A070910 _Benoit Cloitre_, May 20 2002